Abstract
The reconstruction method of homogenized cross sections in the direct response matrix method has been developed. In this reconstruction method, homogenized cross sections, which take into consideration the influences of neighboring fuel assemblies, can be reconstructed with the response relationship of incoming neutron partial currents and neutron production rates. Calculations for heterogeneous multi fuel assembly systems were done to verify the developed method. The thermal energy group fuel assembly cell-averaged homogenized cross sections reconstructed by this method agreed with those evaluated by the direct calculation of the whole system using the Monte Carlo method within 0.2%. The effect using the reconstructed fuel assembly cell-averaged homogenized cross sections in a conventional core analysis code using cross sections homogenized in a fuel assembly cell was also investigated. The results obtained showed that the analysis accuracy of k-infinity can be improved by using the cross sections reconstructed by the method. Because almost no influences on the analysis accuracy could be found related to the divided numbers of the surfaces and the angles, and the response relationship with neutron production rates of fuel rods or a fuel assembly cell-averaged neutron production rate, this reconstruction method is applicable to a conventional core analysis code using homogenized cross sections in a fuel assembly cell.
1. Introduction
As nuclear fuel and core designs have become more advanced, a precise treatment of the effects of the intra- and inter-assembly heterogeneities in core analysis has become more important. Several core analysis codes of a fuel rod geometry unit (i.e., pin-by-pin) have been developed recently for this purpose [Citation1, Citation2]. However, in general, these codes require longer computation times than conventional core analysis codes of a fuel assembly geometry unit [Citation3, Citation4]. If the accuracy of such conventional core analysis codes can be improved, they would be very useful for the design calculations, which need huge parameter surveys for example.
The analysis error factor of such conventional core analysis codes is due to the following, except the error factor by analysis method (e.g., diffusion calculation): using the cross sections homogenized in a fuel assembly cell and evaluating the homogenized cross sections by infinite single fuel assembly calculations. One example of a way to reduce the error associated with the former is the discontinuity factor method [Citation5]. To reduce the error associated with the latter, it is necessary to evaluate the homogenized cross sections by multi fuel assembly calculations. However, it is very difficult to take all combinations of fuel assemblies in a core into consideration.
On the other hand, for the purpose of enabling to evaluate various reaction rates in the core analysis method using the direct response matrix method [Citation6–Citation9], the author has developed the reconstruction method of homogenized cross sections. In this reconstruction method, homogenized cross sections, which take into consideration the influences of neighboring fuel assemblies, are reconstructed with the response relationship of the incoming neutron partial currents and the neutron production rates. Therefore, this reconstruction method may also be applicable to a conventional core analysis code using homogenized cross sections in a fuel assembly cell.
This paper describes the reconstruction method of homogenized cross sections, verification results obtained for this reconstruction method, and the effects of using the reconstructed homogenized cross sections in a conventional core analysis code.
2. Reconstruction method of homogenized cross sections
Here, the reconstruction method of homogenized cross sections in the direct response matrix method is described.
2.1. Fuel assembly analysis
In an ordinary infinite single fuel assembly analysis using a Monte Carlo method, the neutron flux of the gth energy group originating in the neutrons entering from the ti
th segment of the si
th surface at the gi
th energy group is evaluated. The segment denoted above means not only the subsurface obtained by dividing the surface, but also the angular segment obtained by dividing the angle at which neutrons cross the surface. The response relationship of an incoming neutron and a neutron flux 
is obtained by normalizing the neutron flux to that per incoming neutron. Moreover, the neutron flux of the gth energy group originating in the neutrons produced in the fuel rod j is evaluated. The response relationship of a produced neutron and a neutron flux Φ
f
j→g
is obtained by normalizing the neutron flux to that per produced neutron.
When the neutron fluxes originating in the neutrons entering from the surface and produced in the fuel rod are evaluated, the cross sections homogenized in the fuel assembly cell by using these neutron fluxes are also evaluated. That is, the cross section of the gth energy group and reaction x homogenized by using the neutron flux originating in the neutrons entering from the ti
th segment of the si
th surface at the gi
th energy group 
, and the cross section of the gth energy group and the reaction x homogenized by using the neutron flux originating in the neutrons produced in the fuel rod j, Σ
f
x,j→g
are evaluated.
2.2. Core analysis
In core analysis by the direct response matrix method, the incoming neutron partial current through the ti
th segment of the si
th surface at the gi
th energy group, 
, and the neutron production rate of the fuel rod j,
Pj
are obtained. By multiplying these by the response relationships mentioned above, the neutron fluxes of the gth energy group originating in the neutrons entering from the ti
th segment of the si
th surface at the gi
th energy group, 
, and originating in the neutrons produced in the fuel rod j, Φ
f
j→g
are obtained as follows:
By averaging the homogenized cross sections mentioned above by using these fluxes, the node-averaged homogenized cross section of the gth energy group and reaction x, Σ x,g can be reconstructed as follows:
In this reconstruction method, homogenized cross sections are reconstructed with the response relationships of incoming neutron partial currents and neutron production rates. Therefore, if the numbers of subsurfaces and angular segments can be one (no division) and the response relationship with neutron production rates can be set to that with a fuel assembly cell-averaged neutron production rate, from the viewpoint of analysis accuracy, this reconstruction method is applicable to a conventional core analysis code using homogenized cross sections in a fuel assembly cell.
3. Verification of reconstruction method
For verification of the reconstruction method of homogenized cross sections, two heterogeneous two-dimensional systems were calculated. As shown in , each was composed of two low-enriched fuel assemblies and two high-enriched fuel assemblies. These assemblies were typical BWR fuel assemblies consisting of an 8×8 fuel rod array and one large central water rod. The assembly averaged enrichment of the low-enriched fuel assemblies was 1.3 wt% and that of the high-enriched fuel assemblies was 3.8 wt%. Each high-enriched fuel assembly had eight gadolinia-containing fuel rods with an average gadolinia density of 7.0 wt%, whereas the low-enriched fuel assemblies had no such fuel rods. The void fraction of the moderator was 0.4 in both types of fuel assemblies. As shown in , one of these heterogeneous systems had no control rods, and the other had a control rod in one of the low-enriched fuel assemblies. The boundary condition of both systems was reflective, so these systems were extremely heterogeneous.
Figure 1 Geometrical configurations of calculated systems
The fuel assembly cell-averaged homogenized macroscopic cross sections were reconstructed by carrying out core analysis by the direct response matrix method. The sub-response matrices of the direct response matrix method were calculated by the fuel assembly analysis code VMONT [Citation10]. The VMONT code is based on a Monte Carlo neutron transport method and uses a multi-group model for the neutron spectrum calculation for which the total number of energy groups is 190. The number of energy groups of the produced sub-response matrices and the reconstructed cross sections was three: 5.53 keV to 10.0 MeV, 0.625 eV to 5.53 keV, and 0.0 to 0.625 eV. The number of tracked neutrons was 2×106. The statistical uncertainty of the neutron infinite multiplication factor was about 0.03% Δk.
The number of subsurfaces for each surface and the number of angular segments for each subsurface were each set to four. The response relationship with neutron production rates was set to that with neutron production rates of fuel rods. Two additional cases were calculated in consideration that this reconstruction method was applied to a conventional core analysis code using the homogenized cross sections in a fuel assembly cell. In the first additional case, the numbers of subsurfaces and angular segments were each changed to one (no division). In the second, the response relationship with the neutron production rates was changed to that with a fuel assembly cell-averaged neutron production rate, and no division was used as in the first additional case. The reference results were obtained by the direct calculation of the whole system with the VMONT code. The number of tracked neutrons was 2×107. The statistical uncertainty of the neutron infinite multiplication factor was about 0.01% Δk.
Fuel assembly cell-averaged homogenized macroscopic capture and fission cross sections in the no control rod system are shown in . For the case in which the numbers of subsurfaces and angular segments were each four, the reconstructed cross sections agreed with the reference results within 0.4%. Except for the values of the first energy group (fast energy group), which were smaller than the other group's values, the differences between the reconstructed and reference values were less than 0.15%. This reconstruction method could well reconstruct fuel assembly cell-averaged homogenized cross sections. For the case in which the surfaces and angles were not divided and the response relationship with the neutron production rates was set to that with neutron production rates of fuel rods, the reconstructed cross sections also agreed with the reference values within about 0.4%. Regarding the values of the third energy group (thermal energy group), the differences between the reconstructed and reference values were less than 0.42%, which was larger than the value of the previous case (0.15%). For the case in which the surfaces and the angles were not divided and the response relationship with the neutron production rates was set to that with a fuel assembly cell-averaged neutron production rate, the differences between the reconstructed and reference values were almost the same as those for the case in which the response relationship with neutron production rates was set to that with the neutron production rates of fuel rods.
Table 1 Comparison of homogenized macroscopic capture and fission cross sections in the no control rod system
offers comparisons of fuel assembly cell-averaged homogenized capture and fission cross sections with the values evaluated by infinite single fuel assembly calculations in the no control rod system. For example, the thermal capture cross section of the high-enriched fuel assemblies in this system changed about 2% from the value evaluated by the infinite single fuel assembly calculation. For the case in which the numbers of subsurfaces and angular segments were each four, the change could be represented with less than 0.2% of difference. For the case in which the surfaces and angles were not divided, the change could be represented with less than about 0.4%. There was almost no difference in the response relationship with the neutron production rates. This reconstruction method could well reconstruct the fuel assembly cell-averaged homogenized cross sections, which take into consideration the influences of neighboring fuel assemblies.
Table 2 Comparison of homogenized macroscopic capture and fission cross sections with values evaluated by infinite single fuel assembly calculation in the no control rod system
Fuel assembly cell-averaged homogenized macroscopic capture and fission cross sections in the control rod inserted system are shown in . For the case in which the numbers of subsurfaces and angular segments were each four, the reconstructed cross sections agreed with the reference results within about 1.6%. In particular, for the values of the thermal energy group (group 3), which were larger than the other group's values, the differences between the reconstructed and reference values were less than 0.2%. For the case in which the surfaces and angles were not divided and the response relationship with neutron production rates was set to that with the neutron production rates of fuel rods, the reconstructed cross sections also agreed with the reference values within about 1.5%. About the values of the thermal energy group, the differences between the reconstructed and reference values were less than 0.57%, which was larger than the value of the previous case (0.2%). For the case in which the surfaces and angles were not divided and the response relationship with neutron production rates was set to that with a fuel assembly cell-averaged neutron production rate, the differences between the reconstructed and the reference values were almost the same as those for the case in which the response relationship with neutron production rates was set to that with neutron production rates of fuel rods.
Table 3 Comparison of homogenized macroscopic capture and fission cross sections in the control rod inserted system
offers comparisons of fuel assembly cell-averaged homogenized capture and fission cross sections with the values evaluated by infinite single fuel assembly calculations in the control rod inserted system. For example, the thermal capture cross section of the high-enriched fuel assembly in this system changed about 1.5% from the value evaluated by the infinite single fuel assembly calculation. For the case in which the numbers of subsurfaces and angular segments were each four, the change could be represented with less than 0.2% of difference. For the case in which the surfaces and angles were not divided, the change could be represented with less than 0.4%. There was almost no difference in the response relationship with the neutron production rates. This reconstruction method could well reconstruct fuel assembly cell-averaged homogenized cross sections, which take into consideration the influences of neighboring fuel assemblies also in the control rod inserted system.
Table 4 Comparison of homogenized macroscopic capture and fission cross sections with values evaluated by infinite single fuel assembly calculation in the control rod inserted system
4. Effect using reconstructed homogenized cross sections
Here, the effect using the reconstructed fuel assembly cell-averaged homogenized cross sections in a conventional core analysis code using cross sections homogenized in a fuel assembly cell is described.
The systems for the calculation were the same systems that were shown in Section 3. The CITATION code [Citation11] was used as a conventional core analysis code using cross sections homogenized in a fuel assembly cell. In core calculations by CITATION, each fuel assembly cell was divided into 4×4, and the number of energy group was three; their structures were the same as shown in Section 3. Six cases for the fuel assembly cell-averaged homogenized macroscopic cross sections were calculated by CITATION as shown below.
| 1. | All cross sections: evaluated by infinite single fuel assembly analysis with the VMONT code (conventional method). | ||||
| 2. | Σ a , Σ f , νΣ f : reconstructed by this reconstruction method for the case in which the numbers of subsurfaces and angular segments were each four (evaluated in Section 3). The other: evaluated by infinite single fuel assembly analysis. | ||||
| 3. | Σ a , Σ f , νΣ f : reconstructed by this reconstruction method for the case in which the surfaces and angles were not divided, and the response relationship with the neutron production rates was set to that with neutron production rates of fuel rods (evaluated in Section 3). The other: evaluated by infinite single fuel assembly analysis. | ||||
| 4. | Σ a , Σ f , νΣ f : reconstructed by this reconstruction method for the case in which the surfaces and angles were not divided, and the response relationship with the neutron production rates was set to that with a fuel assembly cell-averaged neutron production rate (evaluated in Section 3). The other: evaluated by infinite single fuel assembly analysis. | ||||
| 5. | Σ a , Σ f , νΣ f : evaluated by reference calculation. The other: evaluated by infinite single fuel assembly analysis. | ||||
| 6. | All cross sections: evaluated by reference calculation. | ||||
The absorption, fission, and neutron production cross sections were chosen since the effects of reconstruction seemed to be larger than the other cross sections. This reconstruction method is applicable to the cross sections other than these cross sections, namely the scattering cross section and the diffusion coefficient. The reference calculation was the direct calculation of the whole system with the VMONT code.
The calculation results of neutron infinite multiplication factors (k-infinity) in the no control rod system are shown in . In case 1, which used the cross sections evaluated by the infinite single fuel assembly analysis for all cross sections, the difference with the reference was 0.47% Δk. In case 2, which used the cross sections reconstructed by this reconstruction method in which the numbers of subsurfaces and angular segments were each four, for absorption, fission, and neutron production cross sections, the difference with the reference became 0. Furthermore, in cases 3 and 4, which used the cross sections reconstructed for no division of surfaces and angles, the differences were 0.04% Δk. That is, almost no influences on the analysis accuracy could be found regarding the divided numbers of surfaces and angles, and the response relationship with the neutron production rates. In cases 5 and 6, which used the cross sections evaluated by the reference calculation, the differences with the reference were 0, and they were almost same as the differences in cases 2–4. It was almost sufficient to use only absorption, fission, and neutron production cross sections, which take into consideration the influences of neighboring fuel assemblies, because the differences of cases 5 and 6 were the same. The analysis accuracy of k-infinity could be improved by using the cross sections reconstructed by this reconstruction method.
Table 5 Results of neutron infinite multiplication factors (k-infinity) in the no control rod system
shows the calculation results of fuel assembly averaged fission rate distribution in the no control rod system. The differences with the reference were almost the same in all cases. This reconstruction method did not affect the analysis accuracy of fuel assembly averaged fission rate distribution. The analysis error of fuel assembly averaged fission rate distribution is known to originate in the error of the neutron flux distribution by using the fuel assembly cell-averaged homogenized cross sections and the diffusion calculation.
Figure 2 Fuel assembly averaged fission rate distribution in the no control rod system
The calculation results of neutron infinite multiplication factors (k-infinity) in the control rod inserted system are shown in . In case 1, which used the cross sections evaluated by infinite single fuel assembly analysis for all cross sections, the difference with the reference was 0.56% Δk. In case 2, which used the cross sections reconstructed by this reconstruction method in which the numbers of subsurfaces and angular segments were each four, for absorption, fission, and neutron production cross sections, the difference with the reference was 0.15% Δk. Furthermore, in cases 3 and 4, which used the cross sections reconstructed in which the surfaces and angles were not divided, the differences were 0.11% Δk. That is, almost no influences on the analysis accuracy could be found related to the divided numbers of surfaces and angles, and the response relationship with neutron production rates. In case 5, which used the cross sections evaluated by the reference calculation for absorption, fission, and neutron production cross sections, and case 6 for all cross sections, the differences with the reference were 0.21 and 0.31% Δk, respectively. These results mean the following four things. (1) The analysis accuracy of k-infinity could also be improved by using the cross sections reconstructed by this reconstruction method for the control rod inserted system. (2) The difference for k-infinity due to the error of the reconstructed cross sections was 0.06% Δk (cases 2 and 5). (3) The cross sections other than the absorption, fission, and neutron production cross sections, namely the scattering cross section and the diffusion coefficient also influenced analysis accuracy of k-infinity in the control rod inserted system (cases 5 and 6). (4) Even if the “accurate” fuel assembly cell-averaged homogenized cross sections were used, the analysis error of k-infinity did not become 0, because the analysis error by using fuel assembly cell-averaged homogenized cross sections and the diffusion calculation was significant in the control rod inserted system (case 6).
Table 6 Results of neutron infinite multiplication factors (k-infinity) in the control rod inserted system
shows the calculation results of fuel assembly averaged fission rate distribution in the control rod inserted system. The differences with the reference were almost the same in the case which used the cross sections evaluated by the infinite single assembly analysis and the cases which used the cross sections reconstructed by this reconstruction method. This reconstruction method also had almost no effects for the analysis accuracy of fuel assembly averaged fission rate distribution in the control rod inserted system. This analysis error of fuel assembly averaged fission rate distribution is known to originate in the error of the neutron flux distribution by using the fuel assembly cell-averaged homogenized cross sections and the diffusion calculation.
Figure 3 Fuel assembly averaged fission rate distribution in the control rod inserted system
5. Conclusion
The reconstruction method of homogenized cross sections in the direct response matrix method has been developed. In this reconstruction method, homogenized cross sections, which take into consideration the influences of neighboring fuel assemblies, are reconstructed with the response relationship of the incoming neutron partial currents and the neutron production rates. Therefore, this reconstruction method may be applicable also to the conventional core analysis code using the homogenized cross section in a fuel assembly cell.
For the purpose of verification of the reconstructed method of homogenized cross sections, the calculations for two heterogeneous systems were performed. The cross sections reconstructed by this reconstructed method agreed with those evaluated by the direct calculation of the whole system using the Monte Carlo method within 0.4% in the no control rod system and 1.6% in the control rod inserted system. In particular, regarding the cross sections of the thermal energy group, the differences between the reconstructed and the directly evaluated cross sections were less than 0.15% in the no control rod system and 0.2% in the control rod inserted system. These good agreements mean that this reconstruction method can well reconstruct fuel assembly cell-averaged homogenized cross sections, which take into consideration the influences of neighboring fuel assemblies.
The effect using the reconstructed fuel assembly cell-averaged homogenized cross sections in a conventional core analysis code using cross sections homogenized in a fuel assembly cell was investigated. By using the cross sections reconstructed by this reconstruction method, the differences of k-infinity with the reference (the direct calculation of the whole system using the Monte Carlo method) were reduced from 0.47% Δk to 0% Δk in the no control rod system, and from 0.56% Δk to 0.15% Δk in the control rod inserted system. These results mean that the analysis accuracy of k-infinity can be improved by using the cross sections reconstructed by this reconstruction method. Regarding the fuel assembly averaged fission rate distribution, the differences with the reference were almost the same in the cases which used the cross sections evaluated by the infinite single fuel assembly analysis and reconstructed by this the reconstruction method. These results mean that this reconstruction method has almost no effects for the analysis accuracy of fuel assembly averaged fission rate distribution. This analysis error of fuel assembly averaged fission rate distribution originates in the error of the neutron flux distribution by using the fuel assembly cell-averaged homogenized cross sections and the diffusion calculation. Because almost no influences on the analysis accuracy could be found related to the divided numbers of surfaces and angles, and the response relationship with neutron production rates, this reconstruction method is applicable to a conventional core analysis code using homogenized cross sections in a fuel assembly cell.
Notes
aDivided numbers of surfaces and angles.
bResponse relationship with neutron production rate.
Note: Values in () indicates difference from reference (%).
aDivided numbers of surfaces and angles.
bResponse relationship with neutron production rate.
Note: Values in () indicates difference from values evaluated by infinite single fuel assembly calculation (%).
aDivided numbers of surfaces and angles.
bResponse relationship with neutron production rate.
Note: Values in () indicates difference from reference (%).
aDivided numbers of surfaces and angles.
bResponse relationship with neutron production rate.
Note: Values in () indicates difference from values evaluated by infinite single fuel assembly calculation (%).
aResponse relationship with neutron production rate.
bReference calculation (direct calculation of whole system).
aResponse relationship with neutron production rate.
bReference calculation (direct calculation of whole system).
References
- Tatsumi , M and Yamamoto , A . 2003 . Advanced PWR core calculation based on multi-group nodal-transport method in three-dimensional pin-by-pin geometry . J Nucl Sci Technol , 40 : 376 – 387 . doi: 10.1080/18811248.2003.9715369
- Tada , K , Yamamoto , A , Watanabe , M , Noda , H , Kitamura , Y and Yamane , Y . 2007 . Applicability of the diffusion and simplified P3 theories for BWR pin-by-pin core analysis . Proceedings of the 15th International Conference on Nuclear Engineering . Apr 22–26 2007 , Nagoya , Japan. (ICONE-15)
- Maruyama , H , Koyama , J , Aoyama , M , Ishii , K , Zukeran , A and Kiguchi , T . 1997 . Development of an advanced core analysis system for boiling water reactor designs . Nucl Technol , 118 : 3 – 13 .
- Iwamoto , T , Tamitani , M and Moore , B . 2001 . Present status of GNF new nodal simulator . Trans Am Nucl Soc , 84 : 55
- Smith , K S . 1980 . Spatial homogenization methods for light water reactor analysis [dissertation] , Cambridge , MA : Massachusetts Institute of technology, Department of Nuclear Engineering .
- Moriwaki , M , Ishii , K , Maruyama , H and Aoyama , M . 1999 . A new direct calculation method of response matrices using a Monte Carlo calculation . J Nucl Sci Technol , 36 : 877 – 887 . doi: 10.1080/18811248.1999.9726278
- Ishii , K , Hino , T , Mitsuyasu , T and Aoyama , M . 2009 . Three-dimensional direct response matrix method using a Monte Carlo calculation . J Nucl Sci Technol , 46 : 259 – 267 . doi: 10.1080/18811248.2007.9711529
- Hino , T , Ishii , K , Mitsuyasu , T and Aoyama , M . 2010 . BWR core simulator using three-dimensional direct response matrix and analysis of cold critical experiments . J Nucl Sci Technol , 47 : 482 – 491 . doi: 10.1080/18811248.2010.9711639
- Ishii , K , Mitsuyasu , T , Hino , T and Aoyama , M . 2012 . Analysis of the MOX critical experiments BASALA by pin-by-pin core analysis method using three-dimensional direct response matrix . J Nucl Sci Technol , 49 : 230 – 243 . doi: 10.1080/00223131.2011.649078
- Morimoto , Y , Maruyama , H , Ishii , K and Aoyama , M . 1989 . Neutronic analysis code for fuel assembly using a vectorized Monte Carlo method . Nucl Sci Eng , 103 : 351 – 358 .
- Fowler , T B , Vondy , D R and Cummingham , G W . 1969 . Nuclear reactor core analysis code: CITATION , Oak Ridge , TN : Oak Ridge National Laboratory . (ORNL-TM-2496)